Lesson 10: Converting Repeating Decimals to Fractions

[Pages:6]NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 10 8?7

Lesson 10: Converting Repeating Decimals to Fractions

Classwork Example 1

There is a fraction with an infinite decimal expansion of 0. 81. Find the fraction.

Exercises 1?2

1. There is a fraction with an infinite decimal expansion of 0. 123. Let = 0. 123. a. Explain why looking at 1000 helps us find the fractional representation of .

Lesson 10:

Converting Repeating Decimals to Fractions

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NYS COMMON CORE MATHEMATICS CURRICULUM b. What is as a fraction?

Lesson 10 8?7

c. Is your answer reasonable? Check your answer using a calculator. 2. There is a fraction with a decimal expansion of 0. 4. Find the fraction, and check your answer using a calculator.

Lesson 10:

Converting Repeating Decimals to Fractions

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NYS COMMON CORE MATHEMATICS CURRICULUM

Example 2

Could it be that 2.138 is also a fraction?

Lesson 10 8?7

Lesson 10:

Converting Repeating Decimals to Fractions

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NYS COMMON CORE MATHEMATICS CURRICULUM

Exercises 3?4

3. Find the fraction equal to 1.623. Check your answer using a calculator.

Lesson 10 8?7

4. Find the fraction equal to 2.960. Check your answer using a calculator.

Lesson 10:

Converting Repeating Decimals to Fractions

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 10 8?7

Lesson Summary

Every decimal with a repeating pattern is a rational number, and we have the means to determine the fraction that has a given repeating decimal expansion.

Example: Find the fraction that is equal to the number 0. 567.

Let represent the infinite decimal 0. 567.

= 0. 567 103 = 103(0. 567) 1000 = 567. 567 1000 = 567 + 0. 567 1000 = 567 + 1000 - = 567 + - 999 = 567 999 567 999 = 999

567 63 = 999 = 111

Multiply by 103 because there are 3 digits that repeat. Simplify By addition By substitution; = 0. 567 Subtraction property of equality Simplify Division property of equality

Simplify

This process may need to be used more than once when the repeating digits, as in numbers such as 1.26, do not begin immediately after the decimal.

Irrational numbers are numbers that are not rational. They have infinite decimal expansions that do not repeat and

they cannot be expressed as for integers and with 0.

Problem Set

1. a. Let = 0. 631. Explain why multiplying both sides of this equation by 103 will help us determine the fractional representation of . b. What fraction is ? c. Is your answer reasonable? Check your answer using a calculator.

2. Find the fraction equal to 3.408. Check your answer using a calculator.

3. Find the fraction equal to 0. 5923. Check your answer using a calculator.

4. Find the fraction equal to 2.382. Check your answer using a calculator.

Lesson 10:

Converting Repeating Decimals to Fractions

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NYS COMMON CORE MATHEMATICS CURRICULUM

Lesson 10 8?7

5. Find the fraction equal to 0. 714285. Check your answer using a calculator.

6. Explain why an infinite decimal that is not a repeating decimal cannot be rational.

7. In a previous lesson, we were convinced that it is acceptable to write 0. 9 = 1. Use what you learned today to show that it is true.

8. Examine the following repeating infinite decimals and their fraction equivalents. What do you notice? Why do you think what you observed is true?

0.

81

=

81 99

0.

4

=

4 9

0.

123

=

123 999

0.

60

=

60 99

0.

4311

=

4311 9999

0.

01

=

1 99

0.

3

=

1 3

=

3 9

0. 9 = 1.0

Lesson 10:

Converting Repeating Decimals to Fractions

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